Two-Body Systems

Two-Body Force A two-body system can be defined with internal and external forces. Center of mass R Equal external force Add to get the CM motion Subtract for relative motion F2int m2 r = r1 – r2 F2ext m1 R r2 F1int r1 F1ext

Reduced Mass The internal forces are equal and opposite. Express the equation in terms of a reduced mass m. m less than either m1, m2 m approximately equals the smaller mass when the other is large. for

Central Force The internal force can be expressed in other coordinates. Spherical coordinates Generalized force A force between two bodies can only depend on r. Central force

Kinetic Energy The kinetic energy can be expressed in spherical coordinates. Use reduced mass Lagrange’s equations can be written for a central force. Central force need not be from a potential.

Coordinate Reduction T doesn’t depend on f directly. The angular momentum about the polar axis is constant. Planar motion Include the polar axis in the plane This leaves two coordinates. r, q constant

Angular Momentum T also doesn’t depend on q directly. constant Constant angular momentum Angular momentum J to avoid confusion with the Lagrangian constant

Central Motion Central motion takes place in a plane. Force, velocity, and radius are coplanar Orbital angular momentum is constant. If the central force is time-independent, the orbit is symmetrical about an apse. Apse is where velocity is perpendicular to radius

Central Potential The central force can derive from a potential. Rewrite as differential equation with angular momentum. Central forces have an equivalent Lagrangian.

Time Independence Change the time derivative to an angle derivative. Combine with the equation of motion. The resulting equation describes a trajectory.

Orbit Equation The solution to the differential equation for the trajectory gives the general orbit equation. Let u = 1/r

Inverse Square Force The inverse square force is central. k < 0 for attractive force Choose constant of integration so V() = 0. F2int m2 r = r1 – r2 m1 R r2 F1int r1

Kepler Lagrangian The inverse square Lagrangian can be expressed in polar coordinates. L is independent of time. The total energy is a constant of the motion. Orbit is symmetrical about an apse.

Kepler Orbits The right side of the orbit equation is constant. Equation is integrable Integration constants: e, q0 e related to initial energy Phase angle corresponds to orientation. The substitution can be reversed to get polar or Cartesian coordinates.

Conic Sections The orbit equation describes a conic section. q0 init orientation (set to 0) s is the directrix. The constant e is the eccentricity. sets the shape e < 1 ellipse e =1 parabola e >1 hyperbola r q s focus

Apsidal Position Elliptical orbits have stable apses. Kepler’s first law Minimum and maximum values of r Other orbits only have a minimum The energy is related to e: Set r = r2, no velocity r r1 q r2 s

Angular Momentum The change in area between orbit and focus is dA/dt Related to angular velocity The change is constant due to constant angular momentum. This is Kepler’s 2nd law dr r

Period and Ellipse The area for the whole ellipse relates to the period. semimajor axis: a=(r1+r2)/2. This is Kepler’s 3rd law. Relation holds for all orbits Constant depends on m, k r r1 q r2 s

Effective Potential The problem can be treated in one dimension only. Just radial r term. Minimum in potential implies bounded orbits. For k > 0, no minimum For E > 0, unbounded Veff Veff r r possibly bounded unbounded

Star Systems Star systems can involve both single and multiple stars. Binary stars are a case of a two-body central force problem. Star systems within 10 Pc have been cataloged by RECONS (Jan 2012). Total systems 259 Singles 185 Doubles 55 Triples 15 Quadruples 3 Quintuples 1

Visual Binaries Visual binaries occur when the centers are separated by more than 1”. Atmospheric effects Apparent binaries occur when two stars are near the same coordinates but not close in space.

Binary Mass Kepler’s third law can be made unitless compared to the sun. Mass in solar masses Period in years Semimajor axis in AU Semimajor axis depends on knowing the distance and tilt. Separate masses come from observing the center.

Spectroscopic Binaries Binary systems that are too close require spectroscopy. Doppler shifted lines Velocity measurements

Eclipsing Binaries An orbit inclination of nearly 90° to the observer produces an eclipsing binary. Light levels are used to measure period and radii.